Construction of solutions to the L2-critical KdV equation with a given asymptotic behaviour

Abstract

We consider the critical Korteweg–de Vries (KdV) equation: ${\displaystyle u_t+(u_{\mathrm{xx}}+u^5{)}_x=0,t,x\in \mathbb{R}.}$ Let $R_j(t,x)=Q_{c_j}(x-x_j-c_{j}t)$ ($j=1,\dots ,N$) be $N$ soliton solutions to this equation. Denote $U(t)$ the KdV linear group, and let $V\in H^1$ be with sufficient decay on the right; that is, let $(1+x_{+}^{2+{\delta }_0})V\in L^2$ be for some ${\delta }_0>0$.

We construct a solution $u(t)$ to the critical KdV equation such that ${\displaystyle \underset{t\to \infty }{lim}\Vert u(t)-U(t)V-\sum _{j=1}^N{R}_j(t){\Vert }_{H^1}=0.}$